Memoirs of the Faculty of Engineering,Okayama University,Vol.24, No.2, pp.49-65, March 1990

Characteristics of Errors in Open and Closed Trilateration Nets

Chuji MORI* and Ken-ichi MACHIDA**

(Received January 17,1990)

SYNOPSIS

Distance measurements have been more and more easy

and accurate to carry out, and it is expected that

distance mesurements may provide rather accurate

results than angle measurements. Under these

circumstances, caracteri tics of errors in typical

trilateration nets are investigated. The nets

investigated are as follows: From single row of

chains to pranimetrically extended nets in figure, open

and closed networks with respect to external

constraint, and with and without as to internal

constraint. Computations are performed by use of the

method of condition equations, and behaviours of error

propagation and errors of coordinates of stations in

the nets are shown in case of typical nets. For

example, effects for decrease in error by composing a

double row of chains and by enforcing external

constraints are explained.

1. INTRODUCTION

In civil engineering survey works, surrounding field circumstances

***

Department of Civil Engineering

Sagae Technical High School

49

50 Chuji MORI and Ken~ichi MACH IDA

and requested accuracies differ from job to job, and it is now

possible to adopt various methods and instruments to a control survey

for such a engineering works. Then, profound and integrated

considerations are necessary to select a suitable method for a job in

hand (1 rv 5). Electoromagnetic distance meters have been improved in

this decade and it is expected that new instruments are useful for

small or middle scale control surveying(1,6 N 8).

Among many subjects to be considered, characteristics of errors or

accuracies of trilateration networks are exclusively dealt with in

this paper. The method by condition equations is applied for

adjusting observed data in spite of proposal of some new methods(9,10)

because it was indicated in the previous paper(1) that this method was

useful for inverstigating characteristics of propagation of observed

errors. After foundamental properties of the errors in simple

trilateration chains were inverstigated in that paper, error analyses

of some more different types of survey chain and net have been carried

out.

Various effects of redundant observations, constraint conditions

and pattern or figure of trilateration nets are discussed in view of

an accuracy in this paper. These data will be available for planning

a control survey with small or middle scale.

2. SURVEYING NETS INVESTIGATED

Control survey networks by which the plane coordinates of many new

stations are established, such as consruction survey for public and

private works and location survey of highways, are treated. Main

subjects considered in this paper are caracteristics of errors in

trilateration networks, especially two types of network. The first

ones are open networks which originate at a station of known position

and terminate at a station of unknown position as shown in Fig.1.

The second ones are closed networks which originate at a station of

known position and close on another station of known position as shown

in Fig.4.

Every chain and net extends lengthy to the Y direction rather than

X direction as shown in Figs.1 to 4, and so the scale of a net is

indicated by a number N of trilaterls connected along the Y direction

instead of a total number of trilaterals constructing a whole net.

Any station in the net is called by a number n which indicates a

Errors in Trilaleration Nets 51

x C D B'

V\!VV\IUyA Chain a B Chain h

A" )

Net B, (Net B' )

10000££>Chain I, (Chain i' )

side measured

fixed station(origin)

fixed direction(Y direction)

Chain c

Chain d

o

[Note]

c1SZSZSZ\9·7}J7Chain b

/\lV\tl~?\A Chain e B Net A, (Net A' ) ,

~---..,---........--...----._~- B'

A Chain f

Chain g

[Note]

Chain e : side A,B is measured

Chain g, chain h and Net B : Diagonals are measured

Numerals written on sides show side length in LaChain i' I:origin and I,II:Y direction

Net A' I:origin and I,II:Y direction

Net A" I:origin and I,III:Y direction

Net B' I:origin and I,II:Y direction

Fig.1 Open chains and nets

52 Chuji MORI and Ken·ichi MACIIIDA

distance from the originating station to that station concerned as a

general rue. But, in special case, a number of trilateral elements

connected as far as that station is utilized.

3. ASSUMPTIONS AND METHOD OF COMPUTATION

As the object of this paper is to present caracteristics of errors

in several typical trilateration networks, the simple types of figure

and constraint condition, which are illustrated in Figs. 1 and 4, are

selected. The following assumptions are, moreover, introduced in

order to find out a general view of error propagation and evaluate an

accuracy of a trilatelation project in preliminary stages of a work.

i) The most of sides in the trilateration nets are same length, and

this length is adopted as a standard length of the sides constructing

the nets. Therefore, That length is denoted by LO and is used for a

unit of length. It follows, in the result, that the length of every

side in the nets is unity with a few exceptions.

ii) The standard error 00 of an observation of side length is

constant in spite of the length of a side with a few exceptions and

denoted by EOLO (EO is a dimensionless constant).

iii) Each side length is measured independently.

According to the above assumptions, a propagated error a in an

estimated value is expressed by use of the unit 0 0 in case of lengths

and coordinates, or 00/Lo= EO in case of angles and directions.

Another measure of an accuracy, other than a, is a cofactor Q.

Cofactor Q is frequently used for explaining a pattern of propagation

of observed errors in this paper. The variance 0 2 of an estimated

value is computed according to the equation

QE: 2L 2o 0 ( 1 )

when an observed error 0 0 is known and the cofactor Q is computed.

The cofactors for coordinates of stations are denoted by Q~x and

Qyy , and the following quantity Qpp are used for expressing the

cofactor for planimetric position of a station.

Qpp = Qxx + Qyy ( 2 )

The method of least squares were applied to the trilateration

ETTors in Trilaleration Nets 53

networks illustrated in Figs. 1 and 4 under the above assumptions.

Computational procedure was described in the previous paper. A

process of propagation of the observed length errors to the directions

of the sides and the coordinates of the stations was found easily by

applying the method of condition equations compared with the method by

observation equations. Another merit is that this method is suitable

for use of an usual personal computer.

4. ERRORS IN TRILATERTION NETS WITHOUT EXTERNAL CONSTRAINT

A local plane rectangular coordinate system is introduced for

adjustment computations as it is illusrated in Figs.1 to 3. The origin

of the coordinate system is chosen at the station of known position

and the Y axis is directed along a line passing through the origin.

4.1 Error Propagation in Two Types of Chains

The caracteristics of errors in trilatelation networks obtained

from the previous paper are as follows. If trilaterals are connected

in a figure of a single chain, Q~Q for the successively connected

stations from the origin increases according to a expression with a

cubic function of n. To reduce QQ~ , it is remarkably effective to

make up double row of single chains, that is to construct a single

hexagonal chain. Though a hexagonal chain has only a small number of

redundant observations, Q~~ for the stations in the chain is well

reduced due to strong constraint. On the contrary, in a single chain,

Qk~ is not so much reduced by redandant observations.

The errors in the single row and double row of chain are compaired

in Fig.2 in terms of standard error ° instead of cofactor Q, in which

the values of errors are computed under the assumption 00=10x10-6LO or

EO=10x10- 6 It is evident from Fig.2 that the standard errors of

angles are not so much different in the both chains, but the errors of

directions of sides of the single row chain successively increase as

the sides are apart from the origin. It resul ts in remarkably large

errors of the coordinate X of stations in that chain. In the double

row chain, on the other hand, these errors are fairly small.

54

x

(al Chain a

Chuji MORI and Ken-ichi MACHlDA

13

9.54 16

(bl Chain v"- -:>l ----'L- ---'''---- .:>I

[Note]

9.54,9.49,etc Errors of side lengths. (unit.:10-6 LOl

4.46"Error of angle. ~ : Error of direction

Numerators: errors of coordinates X. (unit.:10- 6LO)

Denominators: errors of coordinates Y. (unit.:10-6LO)

Fig.2 Errors of side lengths, angles, directions

coordinates of stations in the chain a and

under the assumption ao=10xl0- 6LO

and. ,1 ,

4.2 Influences of Figures of Trilateral Elements

Chains a,b,c and d in Fig.l are different in figure of trilateral

elements. The errors in these chains are represented here in terms

of cofactors instead of standard errors. Example of Q~~, Q~y and

Qpp for the station with a distance SL O from the origin along the Y

axis is summerized in Table 1. The station concerned is shown as

sta tion B in the chain a in Fig.l for example. Meaning of the

cofactors is explained in Eqs. (1) and (2). Table 1 tells us that QQQdecreases markedly as a height of trilateral elements incleases, and

Qyy has, nevertheless, the same value through all chains. These

characteristics do not appear in triangulation chains. These are the

unique characteristics in the trilateral chains, but we must call

attention to the assumption that the observed errors of lengths of

every side are the same regardless of the differences of side lengths.

Errors in Tnlateration Nets

Table 1 Cofactors Qxx,Qyy and Qpp of stationswith distance 5LO in single row chains

Chain a Chain b Chain c Chain dQA~ 127 311 57.3 120xxQ"'''' 5 5 5 5yyQpp 132 316 62.3 125

4.3 Influences of Internal Constraints

55

To investigate the effects for accuracy improvement by providing

internal constraint in trilateration networks, redundant observations

are applied to the several sides which are not necessary to construct

fundamental trilateration nets.

Relations between the values of the cofactors for coordinates of

stations and the conditions of redundant observations are described in

Table 2. The six chains in the cases CD to ® in Table 2 are single row

chains with internal constraint. In these cases, the lengths to

which redundant observations are applied are different in each other

as shown in Table 2. It is, therefore, assumed that the standard

errors of the observed sides are proportional to the square root of

that side length or zero as special examples. The details of these

assumptions are also given in the Table. For comparison with the

previous data, the position of the stations listed in Table 2 is the

same as in Table 1.

Table 2 Cofactors Qxx,Qyy and Qpp of stations with distance 5LO

Case Chain Redundant Observed No.of ~x Q" " Qppyyobservation errors redundancy

a 0 126.7 5.0 1 31 . 7

CD e A,B JQ 1 11 3.3 2.5 115.8

@ a A,B, C 0 ./A,B,JC,D 2 96.7 2.5 99.2

® f A,B' fA,B' 1 125.5 4.3 129.8

CD e A,B 0 1 100.0 0 100.0

® a A,B, C,D 0 2 66.7 0 66.7(6) a A,B, A,D, C,B /A,B,JA,D,JC,B 3 111. 4 2.3 113.7(j) g diagonals 1 5 105.7 4.5 110.2

® i' margenal sides 1 4 46.7 2.8 49.5

56 Chuji MORI and Ken-ichi MACHIDA

It will be evident from Teble 2 that

i) case CD is scarecely effective,

ii) excellent improvement of accuracy can not be expected from

observations of diagonal-type long distances as shown in the cases CDand@,

iii) observations of two long parallel distance are effective a

little as shown in the case 0 and

iv) the case @)and® show that precize measurements of reduundant

long distance are fairly effective.

Another case for providing internal constraints is to compose a

rectangular chain by measuring every diagonal as shown in chains g and

h. The one of the results is described as the case (j) in Table 2.

This chain has five redundant observations, but the accuracy is not so

well improved because the constraint condition is effective only for

connecting the neighbouring two trilaterals.

In Fig.2(b), On the contrary, an example in which observations of

short redundant sides provide an effective constraint for reduceing

the errors of coordinates of stations in a chain is shown. That

chain is a double row chain or hexagonal chain. The case ® in Table 2

is the same chain as shown in Fig.2(b). QxQ in the case ® is less

than a half of that in case(]). The above facts tell us that it is

improtant to provide closing conditions for many successive elements

instead of each neighbouring element.

4.4 Positional Errors of Stations

In order to explane propagation of observed errors to the

coordinates of stations, standard errors Ox and 0y of stations in two

types of net are illustrated in Fig.3. It is found that the pattern

of coordinate errors of stations is simi liar in the two nets, the

double row chain and the quadruple one. For more detail comparison,

in Table 3, cofactors for the two nets are shown by arranging the

corresponding points in a same row. Large differences between the

cofactors for the coordinates of stations of the two types of net are

not found. Therefore, it may be recognized that composing the double

row of simple trilateration chains is considerably effective for

decreasing the postional error of stations, but composing the triple

or quadruple row are not so effective.

Errors in Tn/ateration Nets

(a) Net i

20 00'-_...L..._....I1

O--------'~:=...::....::~L------"'---. y

57

1 2 3

(b) Net A'

Fig.3 Positional errors of stations in two types of open nets

4.5 Index of strength of Figures

The purpose of plane trilatertion networks is to determine

coordinates of each station in the nets rather than to know angles

between sides or lengths of sides. Angles and side lengths are

computed by comparably simple expressions from observed lengths and

moreover they are little correlated with the observed lengths in case

of internal constraint. Then, it results that the errors of them

reveals themselves as fairly simple patterns shown in Fig.2. On the

other hand, as the coordinates of stations are computed by successive

summation functions of angles and lengths, the errors of the

coordinates inclease progressively as the stations are apart from the

origin of the net.

On the base of the above considerations, cofactor Qpp is used for

58 Chuji MORI and Ken~ichi MACH IDA

Table 3 Cofactors of stations in the nets i and A'

Net i Net A'

St. QA" Q"I' Qpp St. QI''' QI'I' Qppxx yy xx yy

2 0 0.9 0.9 2 0 0.9 0.9

3 5.5 1 .8 7.3 3 5.2 1 .6 6.8

4 1 6.1 2.6 18.7 4 1 4 . 1 2.4 16.5

5 34.0 3.5 37.5 - - - -

6 2.1 3.0 5.1 5 2.1 3.0 5.0

7 0.6 2.2 2.7 6 0.5 2.1 2.7

8 2.1 2.8 4.9 7 2.0 2.8 4.8

9 9.5 3.0 12.6 8 8.6 3.0 11 .6

10 23.6 3.4 27.0 9 20.0 3.6 23.6

11 45.7 4.0 49.7 - - - -12 1.6 7.0 8.6 11 1 .4 6.9 8.3

13 1.6 6.9 8.5 12 1 .4 6.8 8.2

14 5.4 7.9 13.3 13 5.0 7.4 12.3

15 16.0 8.7 24.8 14 13.4 7.9 21 .3

16 34.0 9.6 43.6 - - - -20 6.3 24.9 31 .2

21 2.3 24.3 26.6

22 2.3 24.3 26.6

23 6.0 24.7 30.8

evaluating a relative accuracy between the nets in this paper. That

is, we assume that strength of figure of a net can be represented by

the value of Qpp for a station. The station with a distance 4LO from

the origin is selected for comparing the values of Qpp in the nets.

The positions of the stations are illustrated by a solid triangles in

Fig.1 .

The values of Qpp for those stations in typical types of single

chain and net are shown in Table 4. Average of cofactors for

estimated lengths in a net are also described in this Table by

denoting as 0li. 0fl equals to (m-r)/m in which m is a total number

of observations and r means a number of redundant observations.

It is known from Table 4 that

i) as efficiency of redundand observations for reduceing the

posi tional error of stations varies from case to case, r merely

represents a degree of redundancy and is nearly independent of

strength of internal constraint or strength of figure in a net, and

ii) Table 4 may be considered as a summary of the data from Tables

to 3, then it is helpful for getting a knowledge on characteristics of

Errors in Tnlaleralion Nels

accuracy of the nets at a glance.

59

Table 4 Relative

strength

in open

chains

and nets

-Net r Qll Qpp

a 0 1 69.3

b 0 1 157.2

c 0 1 36.3

d 0 1 68.0

e 1 0.950 62.9

f 1 0.955 68.3

g 5 0.808 56.5

h 5 0.808 43.5

i 5 0.875 37.5

i' 5 0.875 27.8

A 16 0.802 31 .3A' 16 0.802 29.8A" 16 0.802 19.8

B 25 0.653 24.9

B' 25 0.653 12.3

Table 5 Q and Qmax in open chains and nets

Net N Q Qmax Net N Q Qmax

a 3 4.8 9.3 h 2 2.2 2.95 11 .5 30.3 4 3.9 7.17 23.4 69.3 6 8.3 19.99 41 .8 132 8 16.3 44.6

10 53.9 175 10 28.7 85.211 68.0 223 i 3 5.2 8.6

b 3 6.6 16.4 5 7.1 13.35 20.8 62.9 7 11.2 27.37 48.2 157 9 17.7 49.79 93.0 316 11 26.8 81 4

10 123 425 i' 3 3.4 5.0c 3 6.9 10.9 5 6.1 13.4

5 10.8 19.3 7 10.5 27.87 16.9 36.3 9 17. 1 49.39 25.3 62.3 11 26.1 79.8

10 30.9 86.4 A 3 5.2 8.6d 3 5.5 10.0 5 11 . 1 25.1

5 11.3 31 .0 7 21 .2 56.27 21 .5 68.0 9 15 7 10?9 37.0 125 A' 3 5.2 8.6

10 45.5 130 5 9.7 17.711 58.5 206 7 15.8 31.2

e 3 4.3 8.0 9 23.5 47 75 10.6 26.8 A" 3 3.4 5.07 21 .0 55.2 5 6.2 10.79 38.2 11 6 7 10.7 20.4

13 93.3 302 9 16.3 32.717 185 620 B 2 2.2 2.9

f 2 2.6 4.5 4 5.4 10.56 16.4 47.8 6 10.6 24.98 31 .2 97.5 8 17.8 46.1

10 53.0 173 B' 2 1 .5 1 .812 83.1 279 4 2.7 4.616 173 605 6 5.4 12.4

g 2 2.4 4.1 8 9.6 25.54 5.8 14.36 13.0 38.38 25.4 80.7

10 44.1 147

4.6 Estimation of Cofactors for Coordinates of Stations

Considerations on cofactors for coordinate of stations in a survey

network are the most important subject to make a plan of a control

survey and, as described in the previons sections, these cofactors

vary strikingly with the figure of the net and internal constraints.

Taking into account this fact, mean values and the maximum values of

Qpp are summerized in Table 5 by use of the notations Q and Qmax'

respectively.

60 Chuji MORI and Ken-ichi MACH IDA

Qxx of any stations on the Y axis in the chain a increases in

accordance with the third order polynomials of a distanse, or the

number of connecting trilaterals, from the origin to that stasion.

It is because the angles of any triraterals are independent of every

observed length in the chain, which will be supposed from the

characteristics of positional errors of any stations in a open

traverse. Every single row chain hase this characteristic. When

effectiveness of internal constraint is weak even if redundant

observations are applied to a single row chain, the tendency of

increase of errors is approximately same as above.

In the chain i and i', the angles and directions are not so

strongly correlated with the lengths of the sides which are far from

them, though they are closely correlated with the lengths of the sides

near themselves. This is the reason why Qxx ' hence Qpp, increases

nearly in accordance with the third order polynomials of a distance

from the origin, or station number n, as shown in Fig.2, Tables 3 and

5.

In the planimetrically spreaded nets such as the nets A and B in

Fig.1, the values Qyy are nearly same as those of Qxx ' and the both

are fairly small. It is an outstanding property of widely extended

nets. We can find, in Table 5, that Qmax also increases approximately

in accordance with the third order polynomials of N, and Q in

accordance with the second order polynomials. On the base of these

tendencies, the cofactors Qmax and Q of larger nets than the computed

ones will be able to be estimated by use of Table 5.

5. ERRORS IN TRILATELATION NETS WITH EXTERNAL CONSTRAINT

Trilatertion networks illustrated in Fig.4 are selected as typical

examples of network with external constraint, all of which originate

at a station of known position and close on another station of known

posi tion. These nets have the same figures as shown in Fig.1.

Notation of the nets is refered to Fig.4, that is, subscript 0 is

attached to the notation in Fig.1 for distinguishing.

The one of given stations is chosen as the origin of coordinate

system and the line connecting the two given stations is chosen as the

Y axis in principle, as illustrated in Fig.4.

Errors in Trilateratian Nets 61

x

Wvvv\-yChain eo

Chain fo

Chain ho

Chain. ,10

Net A'a

[Note]

•

Net Ao

side measured

given station

Net B'o

~ given station(origin of coordinates)

Y direction

Fig.4 Closed chains and nets

5.1 Error Propagation in Several Closed Nets

Errors and cofactors for a single row of trilateral chain eO

composed of 17 elements shown in Fig.5 are computed as an example.

Qpp for the stations in the chain are given in Table 6, in which the

notation of stations is also shown in Fig.5. Qpp is fairly smal,ler

than the one in the open chain e shown in Table 2. Increasing rate

of Qpp in accordance with station number n or distance are very small

62 Chuji MORI and Ken-ichi MACIIIDA

and Qpp increases nearly proportionally to square root of n. From

the fact, It may be readily imagined that even if a long chain with

single row trilaterals is constructed between two given stations, the

maximum error of posi tions of sta tions in the chain is not so large.

The difference in effectiveness of two types of constraint,

iaternal and external, may be recognized distinctly by inspecting the

following simple example. We assume here that every side of

trilateral elements of chain is LO=200m and observed error of the side

length is 00=5mm=25x10- 6Lo. Then We can obtain the following

results. When two given points with a distance 1,800m are connected

by the chain eO which is composed of 17 tri talerals or 17 new

stations, the maximum positional error of stations will be ~

00=29mm=144x10- 6LO according to Table 6. In the cases of the chains

a and e, on the otherhand Table 5 shows that the maximum positional

error attains to J69.3 00=42mm=208x1 0-6LO and J55.2 00=37mm=186x1 0-6 LO '

respectively, even if these chains are composed of 7 trilateral

elements only.

Another example is the case of a net A" O' expanded planimetrically.

Positional errors Ox and 0y' insted of Q, are shown in Fig.6 for

illustrating propagation of observed errors. It is evident that

these positional errors increase with distance from the fixed stations

but are remarkably smaller than those in Net A' illustrated in

Fig.3(b). External constraint is also effective for such a net.

2 4 6

Fig.5 station numbers in the closed chain eO with N=17

Table 6 Qpp of stations in the chain eO

station 1 2 3 4 5 6 7Qpp 7.3 13.8 18.0 23.1 27.6 30.3 33.1

Emrs in Tri/aleralion Nets

y

Fig.6 Positional errors of stations in the closed Net A"O

5.2 Posisional Errors and Influences of ~xternal Constraints

63

Relations between cofactors and number N of trilateral elements

composing the chains are summerized in Table 7. In this Table, Q and

Qmax mean an average and the maximum of Qpp, respectively. As single

row chains eO and f O are very "flexible" as described in Table 2, the

maximum errors in those types of chain are fairly larger than the ones

in other types even if the both end stations of the chains are closed

on given positions.

Table 7 Q and Qmax in closed chains

Ch. N Q Qmax Ch. N Q Qmax

eO 3 2.4 2.5 h O 2 1.7 1 . 7

5 3.2 4.0 4 2.0 2.3

9 6.4 8.8 6 2.7 2.913 11 .9 17.9 8 3.9 4.9

17 20.6 33.1 10 5.6 7.7

f O 6 4.7 6.0 i'O 3 2.1 2.3

8 7.0 9.3 5 2.3 2.8

10 10.0 14.7 7 2.9 3.5

12 14.1 21 .4 9 3.7 4.9

16 26.1 42.2 11 4.7 6.4

64 Chuji MORI and Ken-ichi MACHIDA

Table 8 shows the comparison of the values of Q as well as Qmax in

the nets with two types of constraints, internal and external

constraints. It is evident that splendid efficiency for reducing the

maximum positional errors in a chain is obtained by external

constraint, especially in the case of single trilateral chains. By

comparison between nets AO and A"O which are extented planimetrically,

it is found that external constraint is more effective to AO because

it is constrained through a longer distance.

Table 8 Comparison of Q and Qmaxbetween open and closed nets

Open nets Closed nets

N Net Q Qmax Net Q Qmax

9 e 38.2 116 eO 6.4 8.8

10 f 53.0 173 f O 10.0 14.7

10 h 28.7 85.2 h O 5.6 7.7

9 i' 1 7.1 49.5 i'O 3.7 4.9

7 A 21 .2 56.2 AO 3.1 4.2

7 A" 10.7 20.4 A" 3.1 5.108 B' 9.6 25.5 B' 2.8 5.10

6. CONCLUDING REMARKS

Characteristics of errors in trilateration nets are systematically

explained by use of simple and foundamental figures of net such as

single row, double row and quadruple row, as well as several

constraint conditions. The reason why these subjects are

investigated in this paper is that they are useful for making a new

plan for control survey.

The summerize are as follows:

i) Errors of a single row of chain increase seriously in accordance

with its length. To reduce them it is effective to compose a double

row of chain.

ii) Errors of a single row of chain which is composed of elements

with small hight are fairly greater than the one composed of those

with large hight. This is a different characteristic from the case

of triangution.

iii) Internal contraints enforced to a single row of chain are not so

Envrs in Tnlateratian Nels 65

effective to reduce the error in the chain.

iv) External constraints are useful for reducing the error in any

trilateration chains and nets.

REFERENCES

1) K. Machida and C. Mori : Proc., Japan Soc. of Civil Engineers,

No.401 (1989), pp.51-60.

2) C.Mori : ibid., No.359 (1985), pp.117-126.

3) G.M. Young Bulletin Geodesique, No.114 (1974), pp.349-363.

4) J.R. Baker Jour., Surveying and Mapping Division, Proc., Am.

Soc. of Civil Engineers, Vol.98, No. SU2 (1972), pp.157-166.

5) M. Ruopp : Allgemeine Vermessungs-Nachrichten, 71 Jg. (1971),

S.289-299.

6) Japanease Association of Surveyors : Pricise Control Survey,

rev. ed.(1980), pp.55-66/Appendix pp.10-27.

7) G. Wermann : Allgemeine Vermessungs-Nachrichten, 86 Jg. (1979),

S.265-283/435-446.

8) P.R. Wolf and S.D. Johnson Surveying and Mapping, vo1.34

(1974), pp.337-346.

9) N.F. Danial : Jour., Surveying and Mapping Division, Proc., Am.

Soc. of Civil Engineers, Vol.106, No. SU1 (1980), pp.73-93.

10)N.F. Danial : ibid., Vol.105, No. SU1 (1979), pp.67-83.